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Edge even graceful labeling of a graph  G with  p vertices and  q edges is a bijective  f from the set of edge  E G to the set of positive integers  2,4 , … , 2 q such that all the vertex labels  f ∗ V G , given by  f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where  k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to  r -edge even graceful labeling and strong  r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an  r -edge even graceful graph. Furthermore, the minimum number  r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an  r -edge even graceful labeling was found. Finally, we proved that the even cycle  C 2 n has a strong  2 -edge even graceful labeling when  n is even.

journal of mathematics

Extending of Edge Even Graceful Labeling of Graphs to Strong <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>r</mi> </math>-Edge Even Graceful Labeling

Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling

Extending of Edge Even Graceful Labeling of Graphs to Strong $r$ -Edge Even Graceful Labeling